3.605 \(\int (d+e x)^{5/2} (a+c x^2)^3 \, dx\)

Optimal. Leaf size=204 \[ \frac{2 c^2 (d+e x)^{15/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{8 c^2 d (d+e x)^{13/2} \left (3 a e^2+5 c d^2\right )}{13 e^7}+\frac{6 c (d+e x)^{11/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{4 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^3}{7 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}-\frac{12 c^3 d (d+e x)^{17/2}}{17 e^7} \]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (4*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(9/2))/(3*e^7) + (6*c*(c*d^
2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(13/2))/(13*e
^7) + (2*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(15/2))/(5*e^7) - (12*c^3*d*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d +
e*x)^(19/2))/(19*e^7)

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Rubi [A]  time = 0.102997, 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} \[ \frac{2 c^2 (d+e x)^{15/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{8 c^2 d (d+e x)^{13/2} \left (3 a e^2+5 c d^2\right )}{13 e^7}+\frac{6 c (d+e x)^{11/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{4 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^3}{7 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}-\frac{12 c^3 d (d+e x)^{17/2}}{17 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a + c*x^2)^3,x]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (4*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(9/2))/(3*e^7) + (6*c*(c*d^
2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(13/2))/(13*e
^7) + (2*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(15/2))/(5*e^7) - (12*c^3*d*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d +
e*x)^(19/2))/(19*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)^{5/2} \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3 (d+e x)^{5/2}}{e^6}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{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)^{9/2}}{e^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{11/2}}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{13/2}}{e^6}-\frac{6 c^3 d (d+e x)^{15/2}}{e^6}+\frac{c^3 (d+e x)^{17/2}}{e^6}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^3 (d+e x)^{7/2}}{7 e^7}-\frac{4 c d \left (c d^2+a e^2\right )^2 (d+e x)^{9/2}}{3 e^7}+\frac{6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac{8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{13/2}}{13 e^7}+\frac{2 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{15/2}}{5 e^7}-\frac{12 c^3 d (d+e x)^{17/2}}{17 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}\\ \end{align*}

Mathematica [A]  time = 0.254191, size = 188, normalized size = 0.92 \[ \frac{2 \left (\frac{1}{5} c^2 (d+e x)^{15/2} \left (a e^2+5 c d^2\right )-\frac{4}{13} c^2 d (d+e x)^{13/2} \left (3 a e^2+5 c d^2\right )+\frac{3}{11} c (d+e x)^{11/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )-\frac{2}{3} c d (d+e x)^{9/2} \left (a e^2+c d^2\right )^2+\frac{1}{7} (d+e x)^{7/2} \left (a e^2+c d^2\right )^3+\frac{1}{19} c^3 (d+e x)^{19/2}-\frac{6}{17} c^3 d (d+e x)^{17/2}\right )}{e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(a + c*x^2)^3,x]

[Out]

(2*(((c*d^2 + a*e^2)^3*(d + e*x)^(7/2))/7 - (2*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(9/2))/3 + (3*c*(c*d^2 + a*e^2)
*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/11 - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(13/2))/13 + (c^2*(5*c*d^2 +
a*e^2)*(d + e*x)^(15/2))/5 - (6*c^3*d*(d + e*x)^(17/2))/17 + (c^3*(d + e*x)^(19/2))/19))/e^7

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Maple [A]  time = 0.044, size = 205, normalized size = 1. \begin{align*}{\frac{510510\,{c}^{3}{x}^{6}{e}^{6}-360360\,{c}^{3}d{x}^{5}{e}^{5}+1939938\,a{c}^{2}{e}^{6}{x}^{4}+240240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-1193808\,a{c}^{2}d{e}^{5}{x}^{3}-147840\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+2645370\,{a}^{2}c{e}^{6}{x}^{2}+651168\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+80640\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-1175720\,{a}^{2}cd{e}^{5}x-289408\,a{c}^{2}{d}^{3}{e}^{3}x-35840\,{c}^{3}{d}^{5}ex+1385670\,{a}^{3}{e}^{6}+335920\,{a}^{2}c{d}^{2}{e}^{4}+82688\,{d}^{4}{e}^{2}a{c}^{2}+10240\,{c}^{3}{d}^{6}}{4849845\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(c*x^2+a)^3,x)

[Out]

2/4849845*(e*x+d)^(7/2)*(255255*c^3*e^6*x^6-180180*c^3*d*e^5*x^5+969969*a*c^2*e^6*x^4+120120*c^3*d^2*e^4*x^4-5
96904*a*c^2*d*e^5*x^3-73920*c^3*d^3*e^3*x^3+1322685*a^2*c*e^6*x^2+325584*a*c^2*d^2*e^4*x^2+40320*c^3*d^4*e^2*x
^2-587860*a^2*c*d*e^5*x-144704*a*c^2*d^3*e^3*x-17920*c^3*d^5*e*x+692835*a^3*e^6+167960*a^2*c*d^2*e^4+41344*a*c
^2*d^4*e^2+5120*c^3*d^6)/e^7

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Maxima [A]  time = 1.18694, size = 282, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (255255 \,{\left (e x + d\right )}^{\frac{19}{2}} c^{3} - 1711710 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{3} d + 969969 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 1492260 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\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{11}{2}} - 3233230 \,{\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{9}{2}} + 692835 \,{\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{7}{2}}\right )}}{4849845 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 1711710*(e*x + d)^(17/2)*c^3*d + 969969*(5*c^3*d^2 + a*c^2*e^2)*(e*x
+ d)^(15/2) - 1492260*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(13/2) + 1322685*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^
2*c*e^4)*(e*x + d)^(11/2) - 3233230*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(9/2) + 692835*(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)^(7/2))/e^7

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Fricas [B]  time = 1.82979, size = 863, normalized size = 4.23 \begin{align*} \frac{2 \,{\left (255255 \, c^{3} e^{9} x^{9} + 585585 \, c^{3} d e^{8} x^{8} + 5120 \, c^{3} d^{9} + 41344 \, a c^{2} d^{7} e^{2} + 167960 \, a^{2} c d^{5} e^{4} + 692835 \, a^{3} d^{3} e^{6} + 3003 \,{\left (115 \, c^{3} d^{2} e^{7} + 323 \, a c^{2} e^{9}\right )} x^{7} + 231 \,{\left (5 \, c^{3} d^{3} e^{6} + 10013 \, a c^{2} d e^{8}\right )} x^{6} - 63 \,{\left (20 \, c^{3} d^{4} e^{5} - 22933 \, a c^{2} d^{2} e^{7} - 20995 \, a^{2} c e^{9}\right )} x^{5} + 35 \,{\left (40 \, c^{3} d^{5} e^{4} + 323 \, a c^{2} d^{3} e^{6} + 96577 \, a^{2} c d e^{8}\right )} x^{4} - 5 \,{\left (320 \, c^{3} d^{6} e^{3} + 2584 \, a c^{2} d^{4} e^{5} - 474487 \, a^{2} c d^{2} e^{7} - 138567 \, a^{3} e^{9}\right )} x^{3} + 3 \,{\left (640 \, c^{3} d^{7} e^{2} + 5168 \, a c^{2} d^{5} e^{4} + 20995 \, a^{2} c d^{3} e^{6} + 692835 \, a^{3} d e^{8}\right )} x^{2} -{\left (2560 \, c^{3} d^{8} e + 20672 \, a c^{2} d^{6} e^{3} + 83980 \, a^{2} c d^{4} e^{5} - 2078505 \, a^{3} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{4849845 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

2/4849845*(255255*c^3*e^9*x^9 + 585585*c^3*d*e^8*x^8 + 5120*c^3*d^9 + 41344*a*c^2*d^7*e^2 + 167960*a^2*c*d^5*e
^4 + 692835*a^3*d^3*e^6 + 3003*(115*c^3*d^2*e^7 + 323*a*c^2*e^9)*x^7 + 231*(5*c^3*d^3*e^6 + 10013*a*c^2*d*e^8)
*x^6 - 63*(20*c^3*d^4*e^5 - 22933*a*c^2*d^2*e^7 - 20995*a^2*c*e^9)*x^5 + 35*(40*c^3*d^5*e^4 + 323*a*c^2*d^3*e^
6 + 96577*a^2*c*d*e^8)*x^4 - 5*(320*c^3*d^6*e^3 + 2584*a*c^2*d^4*e^5 - 474487*a^2*c*d^2*e^7 - 138567*a^3*e^9)*
x^3 + 3*(640*c^3*d^7*e^2 + 5168*a*c^2*d^5*e^4 + 20995*a^2*c*d^3*e^6 + 692835*a^3*d*e^8)*x^2 - (2560*c^3*d^8*e
+ 20672*a*c^2*d^6*e^3 + 83980*a^2*c*d^4*e^5 - 2078505*a^3*d^2*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 27.7549, size = 945, normalized size = 4.63 \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)**(5/2)*(c*x**2+a)**3,x)

[Out]

a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**3*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
6*a**2*c*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*a**2*c*d*(-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**2*
c*(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**3 + 6*a*c**2*d**2*(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 + 12*a*c**2*d*(-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 + 6*a*c**2*(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**5 + 2*c**3*d**2*(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 + 4*c**3*d*(-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 - 35*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 + 2*c**3*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/
2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e
*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7

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Giac [B]  time = 1.43203, size = 1129, normalized size = 5.53 \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)^(5/2)*(c*x^2+a)^3,x, algorithm="giac")

[Out]

2/14549535*(415701*(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^2*e^(-2) + 125
97*(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^2*e^(-4) + 323*(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^2*e^(-6) + 4849845*(x*e + d)^(3/2)*a^3*d^2 + 277134*(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*d*e^(-2) + 9690*(693*(x*e + d)^(13/2) - 4
095*(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 - 30
03*(x*e + d)^(3/2)*d^5)*a*c^2*d*e^(-4) + 266*(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 + 15315
3*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*c^3*d*e^(-6) + 1939938*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/
2)*d)*a^3*d + 12597*(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^2*c*e^(-2) + 969*(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)*a*c^2*e^(-4) + 7*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876
*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2
)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*c^3*e^(-6) + 1
38567*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3)*e^(-1)