3.13.64 \(\int \frac {(A+B x) (a+c x^2)^2}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=214 \[ \frac {4 c (d+e x)^{3/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 \sqrt {d+e x}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6 (d+e x)^{3/2}}-\frac {4 c \sqrt {d+e x} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6}-\frac {2 c^2 (d+e x)^{5/2} (5 B d-A e)}{5 e^6}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6} \]

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Rubi [A]  time = 0.09, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {772} \begin {gather*} \frac {4 c (d+e x)^{3/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6}-\frac {4 c \sqrt {d+e x} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 \sqrt {d+e x}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6 (d+e x)^{3/2}}-\frac {2 c^2 (d+e x)^{5/2} (5 B d-A e)}{5 e^6}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(3*e^6*(d + e*x)^(3/2)) - (2*(c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^
2))/(e^6*Sqrt[d + e*x]) - (4*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*Sqrt[d + e*x])/e^6 + (4*c*(5*
B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(3/2))/(3*e^6) - (2*c^2*(5*B*d - A*e)*(d + e*x)^(5/2))/(5*e^6) + (2*B
*c^2*(d + e*x)^(7/2))/(7*e^6)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^{5/2}}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^{3/2}}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 \sqrt {d+e x}}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) \sqrt {d+e x}}{e^5}+\frac {c^2 (-5 B d+A e) (d+e x)^{3/2}}{e^5}+\frac {B c^2 (d+e x)^{5/2}}{e^5}\right ) \, dx\\ &=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^6 \sqrt {d+e x}}-\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) \sqrt {d+e x}}{e^6}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{5/2}}{5 e^6}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 214, normalized size = 1.00 \begin {gather*} \frac {14 A e \left (-5 a^2 e^4+10 a c e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )-10 B \left (7 a^2 e^4 (2 d+3 e x)+14 a c e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )+c^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )}{105 e^6 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*A*e*(-5*a^2*e^4 + 10*a*c*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2
- 8*d*e^3*x^3 + 3*e^4*x^4)) - 10*B*(7*a^2*e^4*(2*d + 3*e*x) + 14*a*c*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 -
e^3*x^3) + c^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)))/(105*e^6*
(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.14, size = 301, normalized size = 1.41 \begin {gather*} \frac {2 \left (-35 a^2 A e^5-105 a^2 B e^4 (d+e x)+35 a^2 B d e^4-70 a A c d^2 e^3+420 a A c d e^3 (d+e x)+210 a A c e^3 (d+e x)^2+70 a B c d^3 e^2-630 a B c d^2 e^2 (d+e x)-630 a B c d e^2 (d+e x)^2+70 a B c e^2 (d+e x)^3-35 A c^2 d^4 e+420 A c^2 d^3 e (d+e x)+630 A c^2 d^2 e (d+e x)^2-140 A c^2 d e (d+e x)^3+21 A c^2 e (d+e x)^4+35 B c^2 d^5-525 B c^2 d^4 (d+e x)-1050 B c^2 d^3 (d+e x)^2+350 B c^2 d^2 (d+e x)^3-105 B c^2 d (d+e x)^4+15 B c^2 (d+e x)^5\right )}{105 e^6 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(35*B*c^2*d^5 - 35*A*c^2*d^4*e + 70*a*B*c*d^3*e^2 - 70*a*A*c*d^2*e^3 + 35*a^2*B*d*e^4 - 35*a^2*A*e^5 - 525*
B*c^2*d^4*(d + e*x) + 420*A*c^2*d^3*e*(d + e*x) - 630*a*B*c*d^2*e^2*(d + e*x) + 420*a*A*c*d*e^3*(d + e*x) - 10
5*a^2*B*e^4*(d + e*x) - 1050*B*c^2*d^3*(d + e*x)^2 + 630*A*c^2*d^2*e*(d + e*x)^2 - 630*a*B*c*d*e^2*(d + e*x)^2
+ 210*a*A*c*e^3*(d + e*x)^2 + 350*B*c^2*d^2*(d + e*x)^3 - 140*A*c^2*d*e*(d + e*x)^3 + 70*a*B*c*e^2*(d + e*x)^
3 - 105*B*c^2*d*(d + e*x)^4 + 21*A*c^2*e*(d + e*x)^4 + 15*B*c^2*(d + e*x)^5))/(105*e^6*(d + e*x)^(3/2))

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fricas [A]  time = 0.40, size = 269, normalized size = 1.26 \begin {gather*} \frac {2 \, {\left (15 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 896 \, A c^{2} d^{4} e - 1120 \, B a c d^{3} e^{2} + 560 \, A a c d^{2} e^{3} - 70 \, B a^{2} d e^{4} - 35 \, A a^{2} e^{5} - 3 \, {\left (10 \, B c^{2} d e^{4} - 7 \, A c^{2} e^{5}\right )} x^{4} + 2 \, {\left (40 \, B c^{2} d^{2} e^{3} - 28 \, A c^{2} d e^{4} + 35 \, B a c e^{5}\right )} x^{3} - 6 \, {\left (80 \, B c^{2} d^{3} e^{2} - 56 \, A c^{2} d^{2} e^{3} + 70 \, B a c d e^{4} - 35 \, A a c e^{5}\right )} x^{2} - 3 \, {\left (640 \, B c^{2} d^{4} e - 448 \, A c^{2} d^{3} e^{2} + 560 \, B a c d^{2} e^{3} - 280 \, A a c d e^{4} + 35 \, B a^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 896*A*c^2*d^4*e - 1120*B*a*c*d^3*e^2 + 560*A*a*c*d^2*e^3 - 70*B*a^2
*d*e^4 - 35*A*a^2*e^5 - 3*(10*B*c^2*d*e^4 - 7*A*c^2*e^5)*x^4 + 2*(40*B*c^2*d^2*e^3 - 28*A*c^2*d*e^4 + 35*B*a*c
*e^5)*x^3 - 6*(80*B*c^2*d^3*e^2 - 56*A*c^2*d^2*e^3 + 70*B*a*c*d*e^4 - 35*A*a*c*e^5)*x^2 - 3*(640*B*c^2*d^4*e -
448*A*c^2*d^3*e^2 + 560*B*a*c*d^2*e^3 - 280*A*a*c*d*e^4 + 35*B*a^2*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x
+ d^2*e^6)

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giac [A]  time = 0.27, size = 319, normalized size = 1.49 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{2} e^{36} - 105 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{2} d e^{36} + 350 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{2} d^{2} e^{36} - 1050 \, \sqrt {x e + d} B c^{2} d^{3} e^{36} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{2} e^{37} - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} d e^{37} + 630 \, \sqrt {x e + d} A c^{2} d^{2} e^{37} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} B a c e^{38} - 630 \, \sqrt {x e + d} B a c d e^{38} + 210 \, \sqrt {x e + d} A a c e^{39}\right )} e^{\left (-42\right )} - \frac {2 \, {\left (15 \, {\left (x e + d\right )} B c^{2} d^{4} - B c^{2} d^{5} - 12 \, {\left (x e + d\right )} A c^{2} d^{3} e + A c^{2} d^{4} e + 18 \, {\left (x e + d\right )} B a c d^{2} e^{2} - 2 \, B a c d^{3} e^{2} - 12 \, {\left (x e + d\right )} A a c d e^{3} + 2 \, A a c d^{2} e^{3} + 3 \, {\left (x e + d\right )} B a^{2} e^{4} - B a^{2} d e^{4} + A a^{2} e^{5}\right )} e^{\left (-6\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(x*e + d)^(7/2)*B*c^2*e^36 - 105*(x*e + d)^(5/2)*B*c^2*d*e^36 + 350*(x*e + d)^(3/2)*B*c^2*d^2*e^36 -
1050*sqrt(x*e + d)*B*c^2*d^3*e^36 + 21*(x*e + d)^(5/2)*A*c^2*e^37 - 140*(x*e + d)^(3/2)*A*c^2*d*e^37 + 630*sq
rt(x*e + d)*A*c^2*d^2*e^37 + 70*(x*e + d)^(3/2)*B*a*c*e^38 - 630*sqrt(x*e + d)*B*a*c*d*e^38 + 210*sqrt(x*e + d
)*A*a*c*e^39)*e^(-42) - 2/3*(15*(x*e + d)*B*c^2*d^4 - B*c^2*d^5 - 12*(x*e + d)*A*c^2*d^3*e + A*c^2*d^4*e + 18*
(x*e + d)*B*a*c*d^2*e^2 - 2*B*a*c*d^3*e^2 - 12*(x*e + d)*A*a*c*d*e^3 + 2*A*a*c*d^2*e^3 + 3*(x*e + d)*B*a^2*e^4
- B*a^2*d*e^4 + A*a^2*e^5)*e^(-6)/(x*e + d)^(3/2)

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maple [A]  time = 0.05, size = 259, normalized size = 1.21 \begin {gather*} -\frac {2 \left (-15 B \,c^{2} x^{5} e^{5}-21 A \,c^{2} e^{5} x^{4}+30 B \,c^{2} d \,e^{4} x^{4}+56 A \,c^{2} d \,e^{4} x^{3}-70 B a c \,e^{5} x^{3}-80 B \,c^{2} d^{2} e^{3} x^{3}-210 A a c \,e^{5} x^{2}-336 A \,c^{2} d^{2} e^{3} x^{2}+420 B a c d \,e^{4} x^{2}+480 B \,c^{2} d^{3} e^{2} x^{2}-840 A a c d \,e^{4} x -1344 A \,c^{2} d^{3} e^{2} x +105 B \,a^{2} e^{5} x +1680 B a c \,d^{2} e^{3} x +1920 B \,c^{2} d^{4} e x +35 A \,a^{2} e^{5}-560 A \,d^{2} a c \,e^{3}-896 A \,c^{2} d^{4} e +70 B \,a^{2} d \,e^{4}+1120 B \,d^{3} a c \,e^{2}+1280 B \,c^{2} d^{5}\right )}{105 \left (e x +d \right )^{\frac {3}{2}} e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105/(e*x+d)^(3/2)*(-15*B*c^2*e^5*x^5-21*A*c^2*e^5*x^4+30*B*c^2*d*e^4*x^4+56*A*c^2*d*e^4*x^3-70*B*a*c*e^5*x^
3-80*B*c^2*d^2*e^3*x^3-210*A*a*c*e^5*x^2-336*A*c^2*d^2*e^3*x^2+420*B*a*c*d*e^4*x^2+480*B*c^2*d^3*e^2*x^2-840*A
*a*c*d*e^4*x-1344*A*c^2*d^3*e^2*x+105*B*a^2*e^5*x+1680*B*a*c*d^2*e^3*x+1920*B*c^2*d^4*e*x+35*A*a^2*e^5-560*A*a
*c*d^2*e^3-896*A*c^2*d^4*e+70*B*a^2*d*e^4+1120*B*a*c*d^3*e^2+1280*B*c^2*d^5)/e^6

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maxima [A]  time = 0.68, size = 254, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{2} - 21 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 70 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 210 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} \sqrt {e x + d}}{e^{5}} + \frac {35 \, {\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5} - 3 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{5}}\right )}}{105 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*((15*(e*x + d)^(7/2)*B*c^2 - 21*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(5/2) + 70*(5*B*c^2*d^2 - 2*A*c^2*d*e +
B*a*c*e^2)*(e*x + d)^(3/2) - 210*(5*B*c^2*d^3 - 3*A*c^2*d^2*e + 3*B*a*c*d*e^2 - A*a*c*e^3)*sqrt(e*x + d))/e^5
+ 35*(B*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5 - 3*(5*B*c^2*d^4 -
4*A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 + B*a^2*e^4)*(e*x + d))/((e*x + d)^(3/2)*e^5))/e

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mupad [B]  time = 1.73, size = 249, normalized size = 1.16 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e+4\,B\,a\,c\,e^2\right )}{3\,e^6}-\frac {\left (d+e\,x\right )\,\left (2\,B\,a^2\,e^4+12\,B\,a\,c\,d^2\,e^2-8\,A\,a\,c\,d\,e^3+10\,B\,c^2\,d^4-8\,A\,c^2\,d^3\,e\right )+\frac {2\,A\,a^2\,e^5}{3}-\frac {2\,B\,c^2\,d^5}{3}-\frac {2\,B\,a^2\,d\,e^4}{3}+\frac {2\,A\,c^2\,d^4\,e}{3}+\frac {4\,A\,a\,c\,d^2\,e^3}{3}-\frac {4\,B\,a\,c\,d^3\,e^2}{3}}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {4\,c\,\sqrt {d+e\,x}\,\left (-5\,B\,c\,d^3+3\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {2\,c^2\,\left (A\,e-5\,B\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(3/2)*(20*B*c^2*d^2 + 4*B*a*c*e^2 - 8*A*c^2*d*e))/(3*e^6) - ((d + e*x)*(2*B*a^2*e^4 + 10*B*c^2*d^4
- 8*A*c^2*d^3*e - 8*A*a*c*d*e^3 + 12*B*a*c*d^2*e^2) + (2*A*a^2*e^5)/3 - (2*B*c^2*d^5)/3 - (2*B*a^2*d*e^4)/3 +
(2*A*c^2*d^4*e)/3 + (4*A*a*c*d^2*e^3)/3 - (4*B*a*c*d^3*e^2)/3)/(e^6*(d + e*x)^(3/2)) + (4*c*(d + e*x)^(1/2)*(A
*a*e^3 - 5*B*c*d^3 - 3*B*a*d*e^2 + 3*A*c*d^2*e))/e^6 + (2*B*c^2*(d + e*x)^(7/2))/(7*e^6) + (2*c^2*(A*e - 5*B*d
)*(d + e*x)^(5/2))/(5*e^6)

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sympy [A]  time = 61.95, size = 231, normalized size = 1.08 \begin {gather*} \frac {2 B c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 A c^{2} e - 10 B c^{2} d\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 8 A c^{2} d e + 4 B a c e^{2} + 20 B c^{2} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (4 A a c e^{3} + 12 A c^{2} d^{2} e - 12 B a c d e^{2} - 20 B c^{2} d^{3}\right )}{e^{6}} - \frac {2 \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6} \sqrt {d + e x}} + \frac {2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{3 e^{6} \left (d + e x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*c**2*(d + e*x)**(7/2)/(7*e**6) + (d + e*x)**(5/2)*(2*A*c**2*e - 10*B*c**2*d)/(5*e**6) + (d + e*x)**(3/2)*(
-8*A*c**2*d*e + 4*B*a*c*e**2 + 20*B*c**2*d**2)/(3*e**6) + sqrt(d + e*x)*(4*A*a*c*e**3 + 12*A*c**2*d**2*e - 12*
B*a*c*d*e**2 - 20*B*c**2*d**3)/e**6 - 2*(a*e**2 + c*d**2)*(-4*A*c*d*e + B*a*e**2 + 5*B*c*d**2)/(e**6*sqrt(d +
e*x)) + 2*(-A*e + B*d)*(a*e**2 + c*d**2)**2/(3*e**6*(d + e*x)**(3/2))

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