3.4.4 \(\int (d+e x) (b x+c x^2)^{5/2} \, dx\) [304]

Optimal. Leaf size=175 \[ \frac {5 b^4 (2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}-\frac {5 b^2 (2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {5 b^6 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{9/2}} \]

[Out]

-5/384*b^2*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^3+1/24*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c^2+1/7*
e*(c*x^2+b*x)^(7/2)/c-5/1024*b^6*(-b*e+2*c*d)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(9/2)+5/1024*b^4*(-b*e+2*
c*d)*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^4

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Rubi [A]
time = 0.05, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {654, 626, 634, 212} \begin {gather*} -\frac {5 b^6 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{9/2}}+\frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac {e \left (b x+c x^2\right )^{7/2}}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^4) - (5*b^2*(2*c*d - b*e)*(b + 2*c*x)*(b*x + c*x^2
)^(3/2))/(384*c^3) + ((2*c*d - b*e)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(24*c^2) + (e*(b*x + c*x^2)^(7/2))/(7*c)
- (5*b^6*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*c^(9/2))

Rule 212

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

Rule 626

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 634

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 654

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

Rubi steps

\begin {align*} \int (d+e x) \left (b x+c x^2\right )^{5/2} \, dx &=\frac {e \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {(2 c d-b e) \int \left (b x+c x^2\right )^{5/2} \, dx}{2 c}\\ &=\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 b^2 (2 c d-b e)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{48 c^2}\\ &=-\frac {5 b^2 (2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 b^4 (2 c d-b e)\right ) \int \sqrt {b x+c x^2} \, dx}{256 c^3}\\ &=\frac {5 b^4 (2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}-\frac {5 b^2 (2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 b^6 (2 c d-b e)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2048 c^4}\\ &=\frac {5 b^4 (2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}-\frac {5 b^2 (2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 b^6 (2 c d-b e)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{1024 c^4}\\ &=\frac {5 b^4 (2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^4}-\frac {5 b^2 (2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (b x+c x^2\right )^{7/2}}{7 c}-\frac {5 b^6 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 185, normalized size = 1.06 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-105 b^6 e+70 b^5 c (3 d+e x)-28 b^4 c^2 x (5 d+2 e x)+16 b^3 c^3 x^2 (7 d+3 e x)+512 c^6 x^5 (7 d+6 e x)+256 b c^5 x^4 (35 d+29 e x)+32 b^2 c^4 x^3 (189 d+148 e x)\right )-\frac {105 b^6 (-2 c d+b e) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{21504 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^6*e + 70*b^5*c*(3*d + e*x) - 28*b^4*c^2*x*(5*d + 2*e*x) + 16*b^3*c^3*x^2*(
7*d + 3*e*x) + 512*c^6*x^5*(7*d + 6*e*x) + 256*b*c^5*x^4*(35*d + 29*e*x) + 32*b^2*c^4*x^3*(189*d + 148*e*x)) -
(105*b^6*(-2*c*d + b*e)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(21504*c^(9/2))

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Maple [A]
time = 0.40, size = 263, normalized size = 1.50

method	result	size
risch	\(-\frac {\left (-3072 c^{6} e \,x^{6}-7424 b \,c^{5} e \,x^{5}-3584 c^{6} d \,x^{5}-4736 b^{2} c^{4} e \,x^{4}-8960 b \,c^{5} d \,x^{4}-48 b^{3} c^{3} e \,x^{3}-6048 b^{2} c^{4} d \,x^{3}+56 b^{4} c^{2} e \,x^{2}-112 b^{3} c^{3} d \,x^{2}-70 b^{5} c e x +140 b^{4} c^{2} d x +105 b^{6} e -210 b^{5} c d \right ) x \left (c x +b \right )}{21504 c^{4} \sqrt {x \left (c x +b \right )}}+\frac {5 b^{7} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) e}{2048 c^{\frac {9}{2}}}-\frac {5 b^{6} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d}{1024 c^{\frac {7}{2}}}\)	\(218\)
default	\(e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )+d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )\)	\(263\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

e*(1/7*(c*x^2+b*x)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*(1/8*(2*c*x+b)*(c*x^2+b*x)^(
3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))
))))+d*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*(1/8*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*
x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2)))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (156) = 312\).
time = 0.28, size = 326, normalized size = 1.86 \begin {gather*} \frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d x + \frac {5 \, \sqrt {c x^{2} + b x} b^{4} d x}{256 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d x}{96 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b^{5} x e}{512 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} x e}{192 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b x e}{12 \, c} - \frac {5 \, 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 {5 \, b^{7} e \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} b^{5} d}{512 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d}{12 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b^{6} e}{1024 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} e}{384 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} e}{24 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} e}{7 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*x^2 + b*x)^(5/2)*d*x + 5/256*sqrt(c*x^2 + b*x)*b^4*d*x/c^2 - 5/96*(c*x^2 + b*x)^(3/2)*b^2*d*x/c - 5/512
*sqrt(c*x^2 + b*x)*b^5*x*e/c^3 + 5/192*(c*x^2 + b*x)^(3/2)*b^3*x*e/c^2 - 1/12*(c*x^2 + b*x)^(5/2)*b*x*e/c - 5/
1024*b^6*d*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 5/2048*b^7*e*log(2*c*x + b + 2*sqrt(c*x^2 +
b*x)*sqrt(c))/c^(9/2) + 5/512*sqrt(c*x^2 + b*x)*b^5*d/c^3 - 5/192*(c*x^2 + b*x)^(3/2)*b^3*d/c^2 + 1/12*(c*x^2
+ b*x)^(5/2)*b*d/c - 5/1024*sqrt(c*x^2 + b*x)*b^6*e/c^4 + 5/384*(c*x^2 + b*x)^(3/2)*b^4*e/c^3 - 1/24*(c*x^2 +
b*x)^(5/2)*b^2*e/c^2 + 1/7*(c*x^2 + b*x)^(7/2)*e/c

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Fricas [A]
time = 1.24, size = 391, normalized size = 2.23 \begin {gather*} \left [-\frac {105 \, {\left (2 \, b^{6} c d - b^{7} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (3584 \, c^{7} d x^{5} + 8960 \, b c^{6} d x^{4} + 6048 \, b^{2} c^{5} d x^{3} + 112 \, b^{3} c^{4} d x^{2} - 140 \, b^{4} c^{3} d x + 210 \, b^{5} c^{2} d + {\left (3072 \, c^{7} x^{6} + 7424 \, b c^{6} x^{5} + 4736 \, b^{2} c^{5} x^{4} + 48 \, b^{3} c^{4} x^{3} - 56 \, b^{4} c^{3} x^{2} + 70 \, b^{5} c^{2} x - 105 \, b^{6} c\right )} e\right )} \sqrt {c x^{2} + b x}}{43008 \, c^{5}}, \frac {105 \, {\left (2 \, b^{6} c d - b^{7} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (3584 \, c^{7} d x^{5} + 8960 \, b c^{6} d x^{4} + 6048 \, b^{2} c^{5} d x^{3} + 112 \, b^{3} c^{4} d x^{2} - 140 \, b^{4} c^{3} d x + 210 \, b^{5} c^{2} d + {\left (3072 \, c^{7} x^{6} + 7424 \, b c^{6} x^{5} + 4736 \, b^{2} c^{5} x^{4} + 48 \, b^{3} c^{4} x^{3} - 56 \, b^{4} c^{3} x^{2} + 70 \, b^{5} c^{2} x - 105 \, b^{6} c\right )} e\right )} \sqrt {c x^{2} + b x}}{21504 \, c^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/43008*(105*(2*b^6*c*d - b^7*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(3584*c^7*d*x^5 +
8960*b*c^6*d*x^4 + 6048*b^2*c^5*d*x^3 + 112*b^3*c^4*d*x^2 - 140*b^4*c^3*d*x + 210*b^5*c^2*d + (3072*c^7*x^6 +
7424*b*c^6*x^5 + 4736*b^2*c^5*x^4 + 48*b^3*c^4*x^3 - 56*b^4*c^3*x^2 + 70*b^5*c^2*x - 105*b^6*c)*e)*sqrt(c*x^2
+ b*x))/c^5, 1/21504*(105*(2*b^6*c*d - b^7*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (3584*c^7*d*
x^5 + 8960*b*c^6*d*x^4 + 6048*b^2*c^5*d*x^3 + 112*b^3*c^4*d*x^2 - 140*b^4*c^3*d*x + 210*b^5*c^2*d + (3072*c^7*
x^6 + 7424*b*c^6*x^5 + 4736*b^2*c^5*x^4 + 48*b^3*c^4*x^3 - 56*b^4*c^3*x^2 + 70*b^5*c^2*x - 105*b^6*c)*e)*sqrt(
c*x^2 + b*x))/c^5]

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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 (d + e x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.41, size = 233, normalized size = 1.33 \begin {gather*} \frac {1}{21504} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, c^{2} x e + \frac {14 \, c^{8} d + 29 \, b c^{7} e}{c^{6}}\right )} x + \frac {70 \, b c^{7} d + 37 \, b^{2} c^{6} e}{c^{6}}\right )} x + \frac {3 \, {\left (126 \, b^{2} c^{6} d + b^{3} c^{5} e\right )}}{c^{6}}\right )} x + \frac {7 \, {\left (2 \, b^{3} c^{5} d - b^{4} c^{4} e\right )}}{c^{6}}\right )} x - \frac {35 \, {\left (2 \, b^{4} c^{4} d - b^{5} c^{3} e\right )}}{c^{6}}\right )} x + \frac {105 \, {\left (2 \, b^{5} c^{3} d - b^{6} c^{2} e\right )}}{c^{6}}\right )} + \frac {5 \, {\left (2 \, b^{6} c d - b^{7} e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21504*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*c^2*x*e + (14*c^8*d + 29*b*c^7*e)/c^6)*x + (70*b*c^7*d + 37*b^2*c
^6*e)/c^6)*x + 3*(126*b^2*c^6*d + b^3*c^5*e)/c^6)*x + 7*(2*b^3*c^5*d - b^4*c^4*e)/c^6)*x - 35*(2*b^4*c^4*d - b
^5*c^3*e)/c^6)*x + 105*(2*b^5*c^3*d - b^6*c^2*e)/c^6) + 5/2048*(2*b^6*c*d - b^7*e)*log(abs(-2*(sqrt(c)*x - sqr
t(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,x^2+b\,x\right )}^{5/2}\,\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)*(d + e*x),x)

[Out]

int((b*x + c*x^2)^(5/2)*(d + e*x), x)

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