3.606 \(\int (d+e x)^{3/2} (a+c x^2)^3 \, dx\)

Optimal. Leaf size=204 \[ \frac{6 c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )}{13 e^7}-\frac{8 c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )}{11 e^7}+\frac{2 c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{12 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^7}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^3}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}-\frac{4 c^3 d (d+e x)^{15/2}}{5 e^7} \]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(5/2))/(5*e^7) - (12*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/(7*e^7) + (2*c*(c*d
^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/(3*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(11/2))/(11*e^
7) + (6*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(13/2))/(13*e^7) - (4*c^3*d*(d + e*x)^(15/2))/(5*e^7) + (2*c^3*(d + e*
x)^(17/2))/(17*e^7)

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Rubi [A]  time = 0.0850612, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {697} \[ \frac{6 c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )}{13 e^7}-\frac{8 c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )}{11 e^7}+\frac{2 c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{12 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^7}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^3}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}-\frac{4 c^3 d (d+e x)^{15/2}}{5 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(a + c*x^2)^3,x]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(5/2))/(5*e^7) - (12*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/(7*e^7) + (2*c*(c*d
^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/(3*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(11/2))/(11*e^
7) + (6*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(13/2))/(13*e^7) - (4*c^3*d*(d + e*x)^(15/2))/(5*e^7) + (2*c^3*(d + e*
x)^(17/2))/(17*e^7)

Rule 697

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

Rubi steps

\begin{align*} \int (d+e x)^{3/2} \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{e^6}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{e^6}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{e^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{e^6}-\frac{6 c^3 d (d+e x)^{13/2}}{e^6}+\frac{c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac{12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac{2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac{8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{11 e^7}+\frac{6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac{4 c^3 d (d+e x)^{15/2}}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}\\ \end{align*}

Mathematica [A]  time = 0.196466, size = 188, normalized size = 0.92 \[ \frac{2 \left (\frac{3}{13} c^2 (d+e x)^{13/2} \left (a e^2+5 c d^2\right )-\frac{4}{11} c^2 d (d+e x)^{11/2} \left (3 a e^2+5 c d^2\right )+\frac{1}{3} c (d+e x)^{9/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )-\frac{6}{7} c d (d+e x)^{7/2} \left (a e^2+c d^2\right )^2+\frac{1}{5} (d+e x)^{5/2} \left (a e^2+c d^2\right )^3+\frac{1}{17} c^3 (d+e x)^{17/2}-\frac{2}{5} c^3 d (d+e x)^{15/2}\right )}{e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)*(a + c*x^2)^3,x]

[Out]

(2*(((c*d^2 + a*e^2)^3*(d + e*x)^(5/2))/5 - (6*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/7 + (c*(c*d^2 + a*e^2)*(
5*c*d^2 + a*e^2)*(d + e*x)^(9/2))/3 - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(11/2))/11 + (3*c^2*(5*c*d^2 + a*
e^2)*(d + e*x)^(13/2))/13 - (2*c^3*d*(d + e*x)^(15/2))/5 + (c^3*(d + e*x)^(17/2))/17))/e^7

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Maple [A]  time = 0.045, size = 205, normalized size = 1. \begin{align*}{\frac{30030\,{c}^{3}{x}^{6}{e}^{6}-24024\,{c}^{3}d{x}^{5}{e}^{5}+117810\,a{c}^{2}{e}^{6}{x}^{4}+18480\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-85680\,a{c}^{2}d{e}^{5}{x}^{3}-13440\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+170170\,{a}^{2}c{e}^{6}{x}^{2}+57120\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+8960\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-97240\,{a}^{2}cd{e}^{5}x-32640\,a{c}^{2}{d}^{3}{e}^{3}x-5120\,{c}^{3}{d}^{5}ex+102102\,{a}^{3}{e}^{6}+38896\,{a}^{2}c{d}^{2}{e}^{4}+13056\,{d}^{4}{e}^{2}a{c}^{2}+2048\,{c}^{3}{d}^{6}}{255255\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(c*x^2+a)^3,x)

[Out]

2/255255*(e*x+d)^(5/2)*(15015*c^3*e^6*x^6-12012*c^3*d*e^5*x^5+58905*a*c^2*e^6*x^4+9240*c^3*d^2*e^4*x^4-42840*a
*c^2*d*e^5*x^3-6720*c^3*d^3*e^3*x^3+85085*a^2*c*e^6*x^2+28560*a*c^2*d^2*e^4*x^2+4480*c^3*d^4*e^2*x^2-48620*a^2
*c*d*e^5*x-16320*a*c^2*d^3*e^3*x-2560*c^3*d^5*e*x+51051*a^3*e^6+19448*a^2*c*d^2*e^4+6528*a*c^2*d^4*e^2+1024*c^
3*d^6)/e^7

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Maxima [A]  time = 1.17442, size = 282, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (15015 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{3} - 102102 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} d + 58905 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 92820 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 218790 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 51051 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{255255 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

2/255255*(15015*(e*x + d)^(17/2)*c^3 - 102102*(e*x + d)^(15/2)*c^3*d + 58905*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)
^(13/2) - 92820*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(11/2) + 85085*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)
*(e*x + d)^(9/2) - 218790*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(7/2) + 51051*(c^3*d^6 + 3*a*c^2
*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(e*x + d)^(5/2))/e^7

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Fricas [A]  time = 1.82975, size = 707, normalized size = 3.47 \begin{align*} \frac{2 \,{\left (15015 \, c^{3} e^{8} x^{8} + 18018 \, c^{3} d e^{7} x^{7} + 1024 \, c^{3} d^{8} + 6528 \, a c^{2} d^{6} e^{2} + 19448 \, a^{2} c d^{4} e^{4} + 51051 \, a^{3} d^{2} e^{6} + 231 \,{\left (c^{3} d^{2} e^{6} + 255 \, a c^{2} e^{8}\right )} x^{6} - 126 \,{\left (2 \, c^{3} d^{3} e^{5} - 595 \, a c^{2} d e^{7}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{4} e^{4} + 51 \, a c^{2} d^{2} e^{6} + 2431 \, a^{2} c e^{8}\right )} x^{4} - 10 \,{\left (32 \, c^{3} d^{5} e^{3} + 204 \, a c^{2} d^{3} e^{5} - 12155 \, a^{2} c d e^{7}\right )} x^{3} + 3 \,{\left (128 \, c^{3} d^{6} e^{2} + 816 \, a c^{2} d^{4} e^{4} + 2431 \, a^{2} c d^{2} e^{6} + 17017 \, a^{3} e^{8}\right )} x^{2} - 2 \,{\left (256 \, c^{3} d^{7} e + 1632 \, a c^{2} d^{5} e^{3} + 4862 \, a^{2} c d^{3} e^{5} - 51051 \, a^{3} d e^{7}\right )} x\right )} \sqrt{e x + d}}{255255 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

2/255255*(15015*c^3*e^8*x^8 + 18018*c^3*d*e^7*x^7 + 1024*c^3*d^8 + 6528*a*c^2*d^6*e^2 + 19448*a^2*c*d^4*e^4 +
51051*a^3*d^2*e^6 + 231*(c^3*d^2*e^6 + 255*a*c^2*e^8)*x^6 - 126*(2*c^3*d^3*e^5 - 595*a*c^2*d*e^7)*x^5 + 35*(8*
c^3*d^4*e^4 + 51*a*c^2*d^2*e^6 + 2431*a^2*c*e^8)*x^4 - 10*(32*c^3*d^5*e^3 + 204*a*c^2*d^3*e^5 - 12155*a^2*c*d*
e^7)*x^3 + 3*(128*c^3*d^6*e^2 + 816*a*c^2*d^4*e^4 + 2431*a^2*c*d^2*e^6 + 17017*a^3*e^8)*x^2 - 2*(256*c^3*d^7*e
+ 1632*a*c^2*d^5*e^3 + 4862*a^2*c*d^3*e^5 - 51051*a^3*d*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 16.6605, size = 564, normalized size = 2.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(c*x**2+a)**3,x)

[Out]

a**3*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**3*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 6*a**2*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3
+ 6*a**2*c*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9
)/e**3 + 6*a*c**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d
+ e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 6*a*c**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10
*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5
+ 2*c**3*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*
x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 2*c**3*(-
d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 3
5*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/
e**7

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Giac [B]  time = 1.35377, size = 675, normalized size = 3.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+a)^3,x, algorithm="giac")

[Out]

2/765765*(21879*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*c*d*e^(-2) + 663*(315
*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e +
d)^(3/2)*d^4)*a*c^2*d*e^(-4) + 17*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*
d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/
2)*d^6)*c^3*d*e^(-6) + 255255*(x*e + d)^(3/2)*a^3*d + 7293*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(
x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^2*c*e^(-2) + 255*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)
*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d
^5)*a*c^2*e^(-4) + 7*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*
(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 -
36465*(x*e + d)^(3/2)*d^7)*c^3*e^(-6) + 51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3)*e^(-1)