3.7.7 \(\int \sqrt {d+e x} (a+c x^2)^3 \, dx\) [607]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 204, normalized size of antiderivative = 1.00, 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 = {711} \begin {gather*} \frac {6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac {8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac {6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(a + c*x^2)^3,x]

[Out]

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

Rule 711

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

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Mathematica [A]
time = 0.08, size = 170, normalized size = 0.83 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 a^3 e^6+1287 a^2 c e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+39 a c^2 e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+c^3 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(15015*a^3*e^6 + 1287*a^2*c*e^4*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 39*a*c^2*e^2*(128*d^4 - 1
92*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + c^3*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2
- 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6)))/(45045*e^7)

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Maple [A]
time = 0.42, size = 269, normalized size = 1.32

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 c^{3} x^{6} e^{6}-2772 c^{3} d \,e^{5} x^{5}+12285 a \,c^{2} e^{6} x^{4}+2520 c^{3} d^{2} e^{4} x^{4}-10920 a \,c^{2} d \,e^{5} x^{3}-2240 c^{3} d^{3} e^{3} x^{3}+19305 a^{2} c \,e^{6} x^{2}+9360 a \,c^{2} d^{2} e^{4} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}-15444 a^{2} c d \,e^{5} x -7488 a \,c^{2} d^{3} e^{3} x -1536 c^{3} d^{5} e x +15015 e^{6} a^{3}+10296 e^{4} d^{2} a^{2} c +4992 d^{4} e^{2} c^{2} a +1024 d^{6} c^{3}\right )}{45045 e^{7}}\)	\(205\)
trager	\(\frac {2 \left (3003 c^{3} e^{7} x^{7}+231 c^{3} d \,e^{6} x^{6}+12285 e^{7} c^{2} a \,x^{5}-252 c^{3} d^{2} e^{5} x^{5}+1365 d \,e^{6} c^{2} a \,x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+3861 a^{2} c d \,e^{6} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x -5148 a^{2} c \,d^{2} e^{5} x -2496 a \,c^{2} d^{4} e^{3} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}+10296 a^{2} c \,d^{3} e^{4}+4992 a \,c^{2} d^{5} e^{2}+1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\)	\(259\)
risch	\(\frac {2 \left (3003 c^{3} e^{7} x^{7}+231 c^{3} d \,e^{6} x^{6}+12285 e^{7} c^{2} a \,x^{5}-252 c^{3} d^{2} e^{5} x^{5}+1365 d \,e^{6} c^{2} a \,x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+3861 a^{2} c d \,e^{6} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x -5148 a^{2} c \,d^{2} e^{5} x -2496 a \,c^{2} d^{4} e^{3} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}+10296 a^{2} c \,d^{3} e^{4}+4992 a \,c^{2} d^{5} e^{2}+1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\)	\(259\)
derivativedivides	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\)	\(269\)
default	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\)	\(269\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/e^7*(1/15*c^3*(e*x+d)^(15/2)-6/13*c^3*d*(e*x+d)^(13/2)+1/11*((a*e^2+c*d^2)*c^2+8*c^3*d^2+c*(2*(a*e^2+c*d^2)*
c+4*d^2*c^2))*(e*x+d)^(11/2)+1/9*(-8*(a*e^2+c*d^2)*c^2*d-2*c*d*(2*(a*e^2+c*d^2)*c+4*d^2*c^2))*(e*x+d)^(9/2)+1/
7*((a*e^2+c*d^2)*(2*(a*e^2+c*d^2)*c+4*d^2*c^2)+8*c^2*d^2*(a*e^2+c*d^2)+c*(a*e^2+c*d^2)^2)*(e*x+d)^(7/2)-6/5*(a
*e^2+c*d^2)^2*c*d*(e*x+d)^(5/2)+1/3*(a*e^2+c*d^2)^3*(e*x+d)^(3/2))

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Maxima [A]
time = 0.31, size = 206, normalized size = 1.01 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, {\left (x e + d\right )}^{\frac {15}{2}} c^{3} - 20790 \, {\left (x e + d\right )}^{\frac {13}{2}} c^{3} d + 12285 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {11}{2}} - 20020 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 19305 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 54054 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 15015 \, {\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 (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(x*e + d)^(15/2)*c^3 - 20790*(x*e + d)^(13/2)*c^3*d + 12285*(5*c^3*d^2 + a*c^2*e^2)*(x*e + d)^(1
1/2) - 20020*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(x*e + d)^(9/2) + 19305*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(x*
e + d)^(7/2) - 54054*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(x*e + d)^(5/2) + 15015*(c^3*d^6 + 3*a*c^2*d^4*
e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(x*e + d)^(3/2))*e^(-7)

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Fricas [A]
time = 1.25, size = 234, normalized size = 1.15 \begin {gather*} -\frac {2}{45045} \, {\left (512 \, c^{3} d^{6} x e - 1024 \, c^{3} d^{7} - 39 \, {\left (77 \, c^{3} x^{7} + 315 \, a c^{2} x^{5} + 495 \, a^{2} c x^{3} + 385 \, a^{3} x\right )} e^{7} - 3 \, {\left (77 \, c^{3} d x^{6} + 455 \, a c^{2} d x^{4} + 1287 \, a^{2} c d x^{2} + 5005 \, a^{3} d\right )} e^{6} + 12 \, {\left (21 \, c^{3} d^{2} x^{5} + 130 \, a c^{2} d^{2} x^{3} + 429 \, a^{2} c d^{2} x\right )} e^{5} - 8 \, {\left (35 \, c^{3} d^{3} x^{4} + 234 \, a c^{2} d^{3} x^{2} + 1287 \, a^{2} c d^{3}\right )} e^{4} + 64 \, {\left (5 \, c^{3} d^{4} x^{3} + 39 \, a c^{2} d^{4} x\right )} e^{3} - 384 \, {\left (c^{3} d^{5} x^{2} + 13 \, a c^{2} d^{5}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-2/45045*(512*c^3*d^6*x*e - 1024*c^3*d^7 - 39*(77*c^3*x^7 + 315*a*c^2*x^5 + 495*a^2*c*x^3 + 385*a^3*x)*e^7 - 3
*(77*c^3*d*x^6 + 455*a*c^2*d*x^4 + 1287*a^2*c*d*x^2 + 5005*a^3*d)*e^6 + 12*(21*c^3*d^2*x^5 + 130*a*c^2*d^2*x^3
+ 429*a^2*c*d^2*x)*e^5 - 8*(35*c^3*d^3*x^4 + 234*a*c^2*d^3*x^2 + 1287*a^2*c*d^3)*e^4 + 64*(5*c^3*d^4*x^3 + 39
*a*c^2*d^4*x)*e^3 - 384*(c^3*d^5*x^2 + 13*a*c^2*d^5)*e^2)*sqrt(x*e + d)*e^(-7)

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Sympy [A]
time = 1.93, size = 265, normalized size = 1.30 \begin {gather*} \frac {2 \left (- \frac {6 c^{3} d \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3*(e*x+d)**(1/2),x)

[Out]

2*(-6*c**3*d*(d + e*x)**(13/2)/(13*e**6) + c**3*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(11/2)*(3*a*c**2*e**2
+ 15*c**3*d**2)/(11*e**6) + (d + e*x)**(9/2)*(-12*a*c**2*d*e**2 - 20*c**3*d**3)/(9*e**6) + (d + e*x)**(7/2)*(
3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 + 15*c**3*d**4)/(7*e**6) + (d + e*x)**(5/2)*(-6*a**2*c*d*e**4 - 12*a*c**2*
d**3*e**2 - 6*c**3*d**5)/(5*e**6) + (d + e*x)**(3/2)*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c*
*3*d**6)/(3*e**6))/e

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (170) = 340\).
time = 3.96, size = 498, normalized size = 2.44 \begin {gather*} \frac {2}{45045} \, {\left (9009 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} c d e^{\left (-2\right )} + 429 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a c^{2} d e^{\left (-4\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{3} d e^{\left (-6\right )} + 3861 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} c e^{\left (-2\right )} + 195 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a c^{2} e^{\left (-4\right )} + 7 \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} c^{3} e^{\left (-6\right )} + 45045 \, \sqrt {x e + d} a^{3} d + 15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d*e^(-2) + 429*(35*(x*e
+ d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4
)*a*c^2*d*e^(-4) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e +
d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c^3*d*e^(-6) + 3
861*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*c*e^(-2) +
195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(
x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*c^2*e^(-4) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d +
12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^
5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^3*e^(-6) + 45045*sqrt(x*e + d)*a^3*d + 15015*((x*e +
d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3)*e^(-1)

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Mupad [B]
time = 0.05, size = 187, normalized size = 0.92 \begin {gather*} \frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(d + e*x)^(1/2),x)

[Out]

((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(11/2))/(11*e^7) + ((d + e*x)^(7/2)*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2
*d^2*e^2))/(7*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7) + (2*(a*e^2 + c*d^2)^3*(d + e*x)^(3/2))/(3*e^7) - ((40*
c^3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(9/2))/(9*e^7) - (12*c^3*d*(d + e*x)^(13/2))/(13*e^7) - (12*c*d*(a*e^2 + c
*d^2)^2*(d + e*x)^(5/2))/(5*e^7)

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