3.11.39 \(\int (A+B x) (d+e x) (b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=264 \[ -\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{6144 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{384 c^3}-\frac {5 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^{11/2}}+\frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^5}-\frac {\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2} \]

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Rubi [A]  time = 0.25, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {779, 612, 620, 206} \begin {gather*} \frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^5}-\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{6144 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{384 c^3}-\frac {5 b^6 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{16384 c^{11/2}}-\frac {\left (b x+c x^2\right )^{7/2} (-16 c (A e+B d)+9 b B e-14 B c e x)}{112 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^2*(32*A
*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*A*c^2*d + 9*b^2*B*
e - 16*b*c*(B*d + A*e))*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B*e - 16*c*(B*d + A*e) - 14*B*c*e*x
)*(b*x + c*x^2)^(7/2))/(112*c^2) - (5*b^6*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*ArcTanh[(Sqrt[c]*x)/Sq
rt[b*x + c*x^2]])/(16384*c^(11/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^{5/2} \, dx &=-\frac {(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {\left (\frac {9}{2} b^2 B e+8 c (2 A c d-b (B d+A e))\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac {\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac {5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {\left (5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right )\right ) \int \sqrt {b x+c x^2} \, dx}{4096 c^4}\\ &=\frac {5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 b^6 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac {5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 b^6 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{16384 c^5}\\ &=\frac {5 b^4 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B e-16 c (B d+A e)-14 B c e x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {5 b^6 \left (32 A c^2 d+9 b^2 B e-16 b c (B d+A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.80, size = 315, normalized size = 1.19 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-210 b^6 c (8 A e+8 B d+3 B e x)+56 b^5 c^2 (20 A (3 d+e x)+B x (20 d+9 e x))-16 b^4 c^3 x (28 A (5 d+2 e x)+B x (56 d+27 e x))+128 b^3 c^4 x^2 (2 A (7 d+3 e x)+3 B x (2 d+e x))+256 b^2 c^5 x^3 (A (378 d+296 e x)+B x (296 d+243 e x))+1024 b c^6 x^4 (4 A (35 d+29 e x)+B x (116 d+99 e x))+2048 c^7 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+945 b^7 B e\right )-\frac {105 b^{11/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (-16 b c (A e+B d)+32 A c^2 d+9 b^2 B e\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{344064 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(945*b^7*B*e - 210*b^6*c*(8*B*d + 8*A*e + 3*B*e*x) + 128*b^3*c^4*x^2*(3*B*x*(2*d +
e*x) + 2*A*(7*d + 3*e*x)) + 2048*c^7*x^5*(4*A*(7*d + 6*e*x) + 3*B*x*(8*d + 7*e*x)) + 56*b^5*c^2*(20*A*(3*d +
e*x) + B*x*(20*d + 9*e*x)) - 16*b^4*c^3*x*(28*A*(5*d + 2*e*x) + B*x*(56*d + 27*e*x)) + 1024*b*c^6*x^4*(4*A*(35
*d + 29*e*x) + B*x*(116*d + 99*e*x)) + 256*b^2*c^5*x^3*(B*x*(296*d + 243*e*x) + A*(378*d + 296*e*x))) - (105*b
^(11/2)*(32*A*c^2*d + 9*b^2*B*e - 16*b*c*(B*d + A*e))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c
*x)/b])))/(344064*c^(11/2))

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IntegrateAlgebraic [A]  time = 1.76, size = 407, normalized size = 1.54 \begin {gather*} \frac {5 \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (-16 A b^7 c e+32 A b^6 c^2 d+9 b^8 B e-16 b^7 B c d\right )}{32768 c^{11/2}}+\frac {\sqrt {b x+c x^2} \left (-1680 A b^6 c e+3360 A b^5 c^2 d+1120 A b^5 c^2 e x-2240 A b^4 c^3 d x-896 A b^4 c^3 e x^2+1792 A b^3 c^4 d x^2+768 A b^3 c^4 e x^3+96768 A b^2 c^5 d x^3+75776 A b^2 c^5 e x^4+143360 A b c^6 d x^4+118784 A b c^6 e x^5+57344 A c^7 d x^5+49152 A c^7 e x^6+945 b^7 B e-1680 b^6 B c d-630 b^6 B c e x+1120 b^5 B c^2 d x+504 b^5 B c^2 e x^2-896 b^4 B c^3 d x^2-432 b^4 B c^3 e x^3+768 b^3 B c^4 d x^3+384 b^3 B c^4 e x^4+75776 b^2 B c^5 d x^4+62208 b^2 B c^5 e x^5+118784 b B c^6 d x^5+101376 b B c^6 e x^6+49152 B c^7 d x^6+43008 B c^7 e x^7\right )}{344064 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[b*x + c*x^2]*(-1680*b^6*B*c*d + 3360*A*b^5*c^2*d + 945*b^7*B*e - 1680*A*b^6*c*e + 1120*b^5*B*c^2*d*x - 2
240*A*b^4*c^3*d*x - 630*b^6*B*c*e*x + 1120*A*b^5*c^2*e*x - 896*b^4*B*c^3*d*x^2 + 1792*A*b^3*c^4*d*x^2 + 504*b^
5*B*c^2*e*x^2 - 896*A*b^4*c^3*e*x^2 + 768*b^3*B*c^4*d*x^3 + 96768*A*b^2*c^5*d*x^3 - 432*b^4*B*c^3*e*x^3 + 768*
A*b^3*c^4*e*x^3 + 75776*b^2*B*c^5*d*x^4 + 143360*A*b*c^6*d*x^4 + 384*b^3*B*c^4*e*x^4 + 75776*A*b^2*c^5*e*x^4 +
118784*b*B*c^6*d*x^5 + 57344*A*c^7*d*x^5 + 62208*b^2*B*c^5*e*x^5 + 118784*A*b*c^6*e*x^5 + 49152*B*c^7*d*x^6 +
101376*b*B*c^6*e*x^6 + 49152*A*c^7*e*x^6 + 43008*B*c^7*e*x^7))/(344064*c^5) + (5*(-16*b^7*B*c*d + 32*A*b^6*c^
2*d + 9*b^8*B*e - 16*A*b^7*c*e)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/(32768*c^(11/2))

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fricas [A]  time = 0.45, size = 802, normalized size = 3.04 \begin {gather*} \left [\frac {105 \, {\left (16 \, {\left (B b^{7} c - 2 \, A b^{6} c^{2}\right )} d - {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (43008 \, B c^{8} e x^{7} + 3072 \, {\left (16 \, B c^{8} d + {\left (33 \, B b c^{7} + 16 \, A c^{8}\right )} e\right )} x^{6} + 256 \, {\left (16 \, {\left (29 \, B b c^{7} + 14 \, A c^{8}\right )} d + {\left (243 \, B b^{2} c^{6} + 464 \, A b c^{7}\right )} e\right )} x^{5} + 128 \, {\left (16 \, {\left (37 \, B b^{2} c^{6} + 70 \, A b c^{7}\right )} d + {\left (3 \, B b^{3} c^{5} + 592 \, A b^{2} c^{6}\right )} e\right )} x^{4} + 48 \, {\left (16 \, {\left (B b^{3} c^{5} + 126 \, A b^{2} c^{6}\right )} d - {\left (9 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} e\right )} x^{3} - 56 \, {\left (16 \, {\left (B b^{4} c^{4} - 2 \, A b^{3} c^{5}\right )} d - {\left (9 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} e\right )} x^{2} - 1680 \, {\left (B b^{6} c^{2} - 2 \, A b^{5} c^{3}\right )} d + 105 \, {\left (9 \, B b^{7} c - 16 \, A b^{6} c^{2}\right )} e + 70 \, {\left (16 \, {\left (B b^{5} c^{3} - 2 \, A b^{4} c^{4}\right )} d - {\left (9 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{688128 \, c^{6}}, -\frac {105 \, {\left (16 \, {\left (B b^{7} c - 2 \, A b^{6} c^{2}\right )} d - {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (43008 \, B c^{8} e x^{7} + 3072 \, {\left (16 \, B c^{8} d + {\left (33 \, B b c^{7} + 16 \, A c^{8}\right )} e\right )} x^{6} + 256 \, {\left (16 \, {\left (29 \, B b c^{7} + 14 \, A c^{8}\right )} d + {\left (243 \, B b^{2} c^{6} + 464 \, A b c^{7}\right )} e\right )} x^{5} + 128 \, {\left (16 \, {\left (37 \, B b^{2} c^{6} + 70 \, A b c^{7}\right )} d + {\left (3 \, B b^{3} c^{5} + 592 \, A b^{2} c^{6}\right )} e\right )} x^{4} + 48 \, {\left (16 \, {\left (B b^{3} c^{5} + 126 \, A b^{2} c^{6}\right )} d - {\left (9 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} e\right )} x^{3} - 56 \, {\left (16 \, {\left (B b^{4} c^{4} - 2 \, A b^{3} c^{5}\right )} d - {\left (9 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} e\right )} x^{2} - 1680 \, {\left (B b^{6} c^{2} - 2 \, A b^{5} c^{3}\right )} d + 105 \, {\left (9 \, B b^{7} c - 16 \, A b^{6} c^{2}\right )} e + 70 \, {\left (16 \, {\left (B b^{5} c^{3} - 2 \, A b^{4} c^{4}\right )} d - {\left (9 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x}}{344064 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/688128*(105*(16*(B*b^7*c - 2*A*b^6*c^2)*d - (9*B*b^8 - 16*A*b^7*c)*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2
+ b*x)*sqrt(c)) + 2*(43008*B*c^8*e*x^7 + 3072*(16*B*c^8*d + (33*B*b*c^7 + 16*A*c^8)*e)*x^6 + 256*(16*(29*B*b*c
^7 + 14*A*c^8)*d + (243*B*b^2*c^6 + 464*A*b*c^7)*e)*x^5 + 128*(16*(37*B*b^2*c^6 + 70*A*b*c^7)*d + (3*B*b^3*c^5
+ 592*A*b^2*c^6)*e)*x^4 + 48*(16*(B*b^3*c^5 + 126*A*b^2*c^6)*d - (9*B*b^4*c^4 - 16*A*b^3*c^5)*e)*x^3 - 56*(16
*(B*b^4*c^4 - 2*A*b^3*c^5)*d - (9*B*b^5*c^3 - 16*A*b^4*c^4)*e)*x^2 - 1680*(B*b^6*c^2 - 2*A*b^5*c^3)*d + 105*(9
*B*b^7*c - 16*A*b^6*c^2)*e + 70*(16*(B*b^5*c^3 - 2*A*b^4*c^4)*d - (9*B*b^6*c^2 - 16*A*b^5*c^3)*e)*x)*sqrt(c*x^
2 + b*x))/c^6, -1/344064*(105*(16*(B*b^7*c - 2*A*b^6*c^2)*d - (9*B*b^8 - 16*A*b^7*c)*e)*sqrt(-c)*arctan(sqrt(c
*x^2 + b*x)*sqrt(-c)/(c*x)) - (43008*B*c^8*e*x^7 + 3072*(16*B*c^8*d + (33*B*b*c^7 + 16*A*c^8)*e)*x^6 + 256*(16
*(29*B*b*c^7 + 14*A*c^8)*d + (243*B*b^2*c^6 + 464*A*b*c^7)*e)*x^5 + 128*(16*(37*B*b^2*c^6 + 70*A*b*c^7)*d + (3
*B*b^3*c^5 + 592*A*b^2*c^6)*e)*x^4 + 48*(16*(B*b^3*c^5 + 126*A*b^2*c^6)*d - (9*B*b^4*c^4 - 16*A*b^3*c^5)*e)*x^
3 - 56*(16*(B*b^4*c^4 - 2*A*b^3*c^5)*d - (9*B*b^5*c^3 - 16*A*b^4*c^4)*e)*x^2 - 1680*(B*b^6*c^2 - 2*A*b^5*c^3)*
d + 105*(9*B*b^7*c - 16*A*b^6*c^2)*e + 70*(16*(B*b^5*c^3 - 2*A*b^4*c^4)*d - (9*B*b^6*c^2 - 16*A*b^5*c^3)*e)*x)
*sqrt(c*x^2 + b*x))/c^6]

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giac [A]  time = 0.27, size = 425, normalized size = 1.61 \begin {gather*} \frac {1}{344064} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, B c^{2} x e + \frac {16 \, B c^{9} d + 33 \, B b c^{8} e + 16 \, A c^{9} e}{c^{7}}\right )} x + \frac {464 \, B b c^{8} d + 224 \, A c^{9} d + 243 \, B b^{2} c^{7} e + 464 \, A b c^{8} e}{c^{7}}\right )} x + \frac {592 \, B b^{2} c^{7} d + 1120 \, A b c^{8} d + 3 \, B b^{3} c^{6} e + 592 \, A b^{2} c^{7} e}{c^{7}}\right )} x + \frac {3 \, {\left (16 \, B b^{3} c^{6} d + 2016 \, A b^{2} c^{7} d - 9 \, B b^{4} c^{5} e + 16 \, A b^{3} c^{6} e\right )}}{c^{7}}\right )} x - \frac {7 \, {\left (16 \, B b^{4} c^{5} d - 32 \, A b^{3} c^{6} d - 9 \, B b^{5} c^{4} e + 16 \, A b^{4} c^{5} e\right )}}{c^{7}}\right )} x + \frac {35 \, {\left (16 \, B b^{5} c^{4} d - 32 \, A b^{4} c^{5} d - 9 \, B b^{6} c^{3} e + 16 \, A b^{5} c^{4} e\right )}}{c^{7}}\right )} x - \frac {105 \, {\left (16 \, B b^{6} c^{3} d - 32 \, A b^{5} c^{4} d - 9 \, B b^{7} c^{2} e + 16 \, A b^{6} c^{3} e\right )}}{c^{7}}\right )} - \frac {5 \, {\left (16 \, B b^{7} c d - 32 \, A b^{6} c^{2} d - 9 \, B b^{8} e + 16 \, A b^{7} c e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*B*c^2*x*e + (16*B*c^9*d + 33*B*b*c^8*e + 16*A*c^9*e)/c^7)*x
+ (464*B*b*c^8*d + 224*A*c^9*d + 243*B*b^2*c^7*e + 464*A*b*c^8*e)/c^7)*x + (592*B*b^2*c^7*d + 1120*A*b*c^8*d +
3*B*b^3*c^6*e + 592*A*b^2*c^7*e)/c^7)*x + 3*(16*B*b^3*c^6*d + 2016*A*b^2*c^7*d - 9*B*b^4*c^5*e + 16*A*b^3*c^6
*e)/c^7)*x - 7*(16*B*b^4*c^5*d - 32*A*b^3*c^6*d - 9*B*b^5*c^4*e + 16*A*b^4*c^5*e)/c^7)*x + 35*(16*B*b^5*c^4*d
- 32*A*b^4*c^5*d - 9*B*b^6*c^3*e + 16*A*b^5*c^4*e)/c^7)*x - 105*(16*B*b^6*c^3*d - 32*A*b^5*c^4*d - 9*B*b^7*c^2
*e + 16*A*b^6*c^3*e)/c^7) - 5/32768*(16*B*b^7*c*d - 32*A*b^6*c^2*d - 9*B*b^8*e + 16*A*b^7*c*e)*log(abs(-2*(sqr
t(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(11/2)

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maple [B]  time = 0.05, size = 716, normalized size = 2.71 \begin {gather*} \frac {5 A \,b^{7} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {9}{2}}}-\frac {5 A \,b^{6} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {7}{2}}}-\frac {45 B \,b^{8} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {11}{2}}}+\frac {5 B \,b^{7} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {9}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{5} e x}{512 c^{3}}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{4} d x}{256 c^{2}}+\frac {45 \sqrt {c \,x^{2}+b x}\, B \,b^{6} e x}{8192 c^{4}}-\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{5} d x}{512 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{6} e}{1024 c^{4}}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{5} d}{512 c^{3}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{3} e x}{192 c^{2}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2} d x}{96 c}+\frac {45 \sqrt {c \,x^{2}+b x}\, B \,b^{7} e}{16384 c^{5}}-\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{6} d}{1024 c^{4}}-\frac {15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{4} e x}{1024 c^{3}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3} d x}{192 c^{2}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{4} e}{384 c^{3}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{3} d}{192 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A b e x}{12 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A d x}{6}-\frac {15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{5} e}{2048 c^{4}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{4} d}{384 c^{3}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{2} e x}{64 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b d x}{12 c}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A \,b^{2} e}{24 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A b d}{12 c}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{3} e}{128 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{2} d}{24 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} B e x}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} A e}{7 c}-\frac {9 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B b e}{112 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} B d}{7 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)*(c*x^2+b*x)^(5/2),x)

[Out]

-15/2048*B*e*b^5/c^4*(c*x^2+b*x)^(3/2)+45/16384*B*e*b^7/c^5*(c*x^2+b*x)^(1/2)+1/6*A*d*x*(c*x^2+b*x)^(5/2)+1/7*
(c*x^2+b*x)^(7/2)/c*B*d+1/7*(c*x^2+b*x)^(7/2)/c*A*e+45/8192*B*e*b^6/c^4*(c*x^2+b*x)^(1/2)*x+3/64*B*e*b^2/c^2*x
*(c*x^2+b*x)^(5/2)-15/1024*B*e*b^4/c^3*(c*x^2+b*x)^(3/2)*x-1/12*b/c*x*(c*x^2+b*x)^(5/2)*B*d+5/192*b^3/c^2*(c*x
^2+b*x)^(3/2)*x*A*e-1/12*b/c*x*(c*x^2+b*x)^(5/2)*A*e-5/96*A*d*b^2/c*(c*x^2+b*x)^(3/2)*x+5/256*A*d*b^4/c^2*(c*x
^2+b*x)^(1/2)*x-5/512*b^5/c^3*(c*x^2+b*x)^(1/2)*x*A*e-5/512*b^5/c^3*(c*x^2+b*x)^(1/2)*x*B*d+5/192*b^3/c^2*(c*x
^2+b*x)^(3/2)*x*B*d+5/384*b^4/c^3*(c*x^2+b*x)^(3/2)*A*e+5/384*b^4/c^3*(c*x^2+b*x)^(3/2)*B*d-1/24*b^2/c^2*(c*x^
2+b*x)^(5/2)*A*e-1/24*b^2/c^2*(c*x^2+b*x)^(5/2)*B*d-5/1024*b^6/c^4*(c*x^2+b*x)^(1/2)*A*e-5/1024*b^6/c^4*(c*x^2
+b*x)^(1/2)*B*d+5/2048*b^7/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*e+1/12*A*d/c*(c*x^2+b*x)^(5/2)*
b-5/192*A*d*b^3/c^2*(c*x^2+b*x)^(3/2)+1/8*B*e*x*(c*x^2+b*x)^(7/2)/c-9/112*B*e*b/c^2*(c*x^2+b*x)^(7/2)+3/128*B*
e*b^3/c^3*(c*x^2+b*x)^(5/2)-5/1024*A*d*b^6/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+5/2048*b^7/c^(9/2
)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d-45/32768*B*e*b^8/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(
1/2))+5/512*A*d*b^5/c^3*(c*x^2+b*x)^(1/2)

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maxima [B]  time = 0.54, size = 573, normalized size = 2.17 \begin {gather*} \frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A d x + \frac {5 \, \sqrt {c x^{2} + b x} A b^{4} d x}{256 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{2} d x}{96 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} B b^{6} e x}{8192 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} e x}{1024 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} e x}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B e x}{8 \, c} - \frac {5 \, A b^{6} d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} - \frac {45 \, B b^{8} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{5} d}{512 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{3} d}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b d}{12 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} B b^{7} e}{16384 \, c^{5}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{5} e}{2048 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{3} e}{128 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b e}{112 \, c^{2}} - \frac {5 \, \sqrt {c x^{2} + b x} {\left (B d + A e\right )} b^{5} x}{512 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B d + A e\right )} b^{3} x}{192 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B d + A e\right )} b x}{12 \, c} + \frac {5 \, {\left (B d + A e\right )} b^{7} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {9}{2}}} - \frac {5 \, \sqrt {c x^{2} + b x} {\left (B d + A e\right )} b^{6}}{1024 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B d + A e\right )} b^{4}}{384 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B d + A e\right )} b^{2}}{24 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} {\left (B d + A e\right )}}{7 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

1/6*(c*x^2 + b*x)^(5/2)*A*d*x + 5/256*sqrt(c*x^2 + b*x)*A*b^4*d*x/c^2 - 5/96*(c*x^2 + b*x)^(3/2)*A*b^2*d*x/c +
45/8192*sqrt(c*x^2 + b*x)*B*b^6*e*x/c^4 - 15/1024*(c*x^2 + b*x)^(3/2)*B*b^4*e*x/c^3 + 3/64*(c*x^2 + b*x)^(5/2
)*B*b^2*e*x/c^2 + 1/8*(c*x^2 + b*x)^(7/2)*B*e*x/c - 5/1024*A*b^6*d*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)
)/c^(7/2) - 45/32768*B*b^8*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) + 5/512*sqrt(c*x^2 + b*x)*A
*b^5*d/c^3 - 5/192*(c*x^2 + b*x)^(3/2)*A*b^3*d/c^2 + 1/12*(c*x^2 + b*x)^(5/2)*A*b*d/c + 45/16384*sqrt(c*x^2 +
b*x)*B*b^7*e/c^5 - 15/2048*(c*x^2 + b*x)^(3/2)*B*b^5*e/c^4 + 3/128*(c*x^2 + b*x)^(5/2)*B*b^3*e/c^3 - 9/112*(c*
x^2 + b*x)^(7/2)*B*b*e/c^2 - 5/512*sqrt(c*x^2 + b*x)*(B*d + A*e)*b^5*x/c^3 + 5/192*(c*x^2 + b*x)^(3/2)*(B*d +
A*e)*b^3*x/c^2 - 1/12*(c*x^2 + b*x)^(5/2)*(B*d + A*e)*b*x/c + 5/2048*(B*d + A*e)*b^7*log(2*c*x + b + 2*sqrt(c*
x^2 + b*x)*sqrt(c))/c^(9/2) - 5/1024*sqrt(c*x^2 + b*x)*(B*d + A*e)*b^6/c^4 + 5/384*(c*x^2 + b*x)^(3/2)*(B*d +
A*e)*b^4/c^3 - 1/24*(c*x^2 + b*x)^(5/2)*(B*d + A*e)*b^2/c^2 + 1/7*(c*x^2 + b*x)^(7/2)*(B*d + A*e)/c

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )\,\left (d+e\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^(5/2)*(A + B*x)*(d + e*x),x)

[Out]

int((b*x + c*x^2)^(5/2)*(A + B*x)*(d + e*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right ) \left (d + e x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)*(d + e*x), x)

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